Algorithms and certificates for exact Mixed Integer Programming

Mixed Integer Programming (MIP) is a powerful tool for solving optimization problems with discrete decisions. Although the problem class of mixed integer programs is NP-hard, MIP solvers have made significant progress in solving large-scale instances through decades of dedicated research and complex algorithmic improvements. In practice, virtually all algorithms to solve MIP problems are based on floating-point arithmetic due to its rapid computation times and robust numerical capabilities. Using error tolerances, MIP solvers avoid numerical issues and maintain a solution quality that is sufficient for most practical applications. There are, however, applications where exact solutions are required, such as when MIPs are employed as a tool in computer-assisted proofs. In such cases, exact, certified MIP solvers are a necessity. This thesis delves into the study and development of exact, certified mixed integer programming methods. The primary contribution of this thesis is an algorithmic framework for exact rational mixed integer programming. This framework incorporates safe dual bounding techniques, exact rational presolving, an exact repair step for heuristic solutions, and a novel exact Gomory mixed integer cut generator. We validate the effectiveness of our framework through experiments on a diverse set of benchmark instances. We take great care to measure the similarities and differences with corresponding techniques in the floating-point setting. Furthermore, we incorporate certification techniques into the framework to provide rigorous guarantees on the correctness of the computed solutions. We also provide a comprehensive algorithmic and computational study of the solver-independent verification of these certificates. We also present a framework for applying exact MIP as a tool for computer-assisted mathematics, using Chvátal’s conjecture as an illustrative example. Another significant contribution of this thesis is a new algorithm for solving linear programs exactly. This algorithm combines two state-of-the-art techniques for exact linear programming: precision boosting and LP iterative refinement. Combining these techniques can significantly improve the performance of exact linear programming solvers, and we prove that the algorithm is theoretically guaranteed to terminate with an exact solution.